July  2017, 37(7): 4035-4051. doi: 10.3934/dcds.2017171

A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian

1. 

Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, 06123 Perugia, Italy

2. 

College of Science, Civil Aviation University of China, Tianjin 300300, China

3. 

Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China

* Corresponding author

Received  March 2016 Revised  February 2017 Published  April 2017

In this paper, we study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we are concerned with the following initial-boundary value problem involving the fractional $p$-Laplacian $\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+M([u]_{s, p}^{p}\text{)}(-\Delta)_{p}^{s}u=f(x, t) & \text{in }\Omega \times {{\mathbb{R}}^{+}}, {{\partial }_{t}}u=\partial u/\partial t, \\ u(x, 0)={{u}_{0}}(x) & \text{in }\Omega, \\ u=0\ & \text{in }{{\mathbb{R}}^{N}}\backslash \Omega, \\\end{array}\text{ }\ \ \right.$ where $[u]_{s, p}$ is the Gagliardo $p$-seminorm of $u$, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partialΩ$, $1 < p < N/s$, with $0 < s < 1$, the main Kirchhoff function $M:\mathbb{R}^{ + }_{0} \to \mathbb{R}^{ + }$ is a continuous and nondecreasing function, $(-Δ)_p^s$ is the fractional $p$-Laplacian, $u_0$ is in $L^2(Ω)$ and $f∈ L^2_{\rm loc}(\mathbb{R}^{ + }_0;L^2(Ω))$. Under some appropriate conditions, the well-posedness of solutions for the problem above is studied by employing the sub-differential approach. Finally, the large-time behavior and extinction of solutions are also investigated.

Citation: Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
References:
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London,, 1975.   Google Scholar
[2]

G. Akagi and K. Matsuura, Well-posedness and large-time behaviors of solutions for a parabolic equations involving $p(x)$-Laplacian, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications, 8th AIMS Conference.Suppl., 1 (2011), 22-31.   Google Scholar

[3]

G. Akagi, K. Matsuura, Nonlinear diffusion equations driven by the $p(·)$-Laplacian, Nonlinear Differential Equations Appl. NoDEA, 20 (2013), 37-64.  doi: 10.1007/s00030-012-0153-6.  Google Scholar

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F. AndreuJ. M. MazónJ. D. Rossi and J. Toledo, A nonlocal $p$-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions, SIAM J. Math. Anal., 40 (2009), 1815-1851.  doi: 10.1137/080720991.  Google Scholar

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S. Antontsev and S. Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math.(234), 2010 (), 2633-2645.  doi: 10.1016/j.cam.2010.01.026.  Google Scholar

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S. Antontsev, S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, J. Math. Anal. Appl., 361 (2010), 371-391.  doi: 10.1016/j.jmaa.2009.07.019.  Google Scholar

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D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.   Google Scholar

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G. AutuoriA. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014.  Google Scholar

[9]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.  Google Scholar

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L. Caffarelli, Some nonlinear problems involving non-local diffusions, ICIAM 07-6th International Congress on Industrial and Applied Mathematics, Eur. Math. Soc., Zürich, (2009), 43-56.  doi: 10.4171/056-1/3.  Google Scholar

[12]

L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

[13]

E. ChasseigneM. Chaves and J. D. Rossi, Asymptotic behaviour for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.  doi: 10.1016/j.matpur.2006.04.005.  Google Scholar

[14]

F. Colasuonno and P. Pucci, Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.  doi: 10.1016/j.na.2011.05.073.  Google Scholar

[15]

C. CortazarM. ElguetaJ. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002.  Google Scholar

[16]

A. Di CastroT. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836.  doi: 10.1016/j.jfa.2014.05.023.  Google Scholar

[17]

A. Di CastroT. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003.  Google Scholar

[18]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[19]

J. M. do'OO. H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1395-1410.  doi: 10.1007/s00030-015-0327-0.  Google Scholar

[20]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.   Google Scholar

[21]

M. Fila, Boundedness of global solutions of nonlinear diffusion equations, J. Differential Equations, 98 (1992), 226-240.  doi: 10.1016/0022-0396(92)90091-Z.  Google Scholar

[22]

A. FiscellaR. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.  doi: 10.5186/aasfm.2015.4009.  Google Scholar

[23]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011.  Google Scholar

[24]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma, 5 (2014), 373-386.   Google Scholar

[25]

M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Meth. Appl. Sci., 22 (1999), 375-388.  doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7.  Google Scholar

[26]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[27]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[28]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.  doi: 10.1007/s00526-013-0600-1.  Google Scholar

[29]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977.  doi: 10.1016/j.na.2005.03.021.  Google Scholar

[30]

X. MingqiG. Molica BisciG. H. Tian and B. L. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 357-374.  doi: 10.1088/0951-7715/29/2/357.  Google Scholar

[31]

M. Pérez-Llanosa and J. D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640.  doi: 10.1016/j.na.2008.02.076.  Google Scholar

[32]

P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.  doi: 10.4171/RMI/879.  Google Scholar

[33]

P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214.  doi: 10.1006/jdeq.1998.3477.  Google Scholar

[34]

P. PucciM. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogenous Schrodinger-Kirchhoff type equations involving the fractional $p-$Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5.  Google Scholar

[35]

P. PucciM. Q. Xiang and B. L. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102.  Google Scholar

[36]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs Vol. 49, American Mathematical Society, Providence, RI, 1997, xiv + 278 pp.  Google Scholar

[37]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symp., Springer, Heidelberg, 7 (2012), 271-298.  doi: 10.1007/978-3-642-25361-4_15.  Google Scholar

[38]

M. Q. XiangB. L. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041.  doi: 10.1016/j.jmaa.2014.11.055.  Google Scholar

[39]

M. Q. XiangB. L. Zhang and M. Ferrara, Multiplicity results for the nonhomogeneous fractional $p$-Kirchhoff equations with concave-convex nonlinearities, Proc. Roy. Soc. A, 471 (2015), 20150034, 14 pp.  doi: 10.1098/rspa.2015.0034.  Google Scholar

[40]

M. Q. XiangB. L. Zhang and V. Rădulescu, Existence of solutions for perturbed fractional $p$-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413.  doi: 10.1016/j.jde.2015.09.028.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic Press, New York-London,, 1975.   Google Scholar
[2]

G. Akagi and K. Matsuura, Well-posedness and large-time behaviors of solutions for a parabolic equations involving $p(x)$-Laplacian, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications, 8th AIMS Conference.Suppl., 1 (2011), 22-31.   Google Scholar

[3]

G. Akagi, K. Matsuura, Nonlinear diffusion equations driven by the $p(·)$-Laplacian, Nonlinear Differential Equations Appl. NoDEA, 20 (2013), 37-64.  doi: 10.1007/s00030-012-0153-6.  Google Scholar

[4]

F. AndreuJ. M. MazónJ. D. Rossi and J. Toledo, A nonlocal $p$-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions, SIAM J. Math. Anal., 40 (2009), 1815-1851.  doi: 10.1137/080720991.  Google Scholar

[5]

S. Antontsev and S. Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math.(234), 2010 (), 2633-2645.  doi: 10.1016/j.cam.2010.01.026.  Google Scholar

[6]

S. Antontsev, S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, J. Math. Anal. Appl., 361 (2010), 371-391.  doi: 10.1016/j.jmaa.2009.07.019.  Google Scholar

[7]

D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.   Google Scholar

[8]

G. AutuoriA. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014.  Google Scholar

[9]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.  Google Scholar

[10] H. Brézis, Operateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, Math Studies, Vol.5 North-Holland, Amsterdam, New York, 1973.   Google Scholar
[11]

L. Caffarelli, Some nonlinear problems involving non-local diffusions, ICIAM 07-6th International Congress on Industrial and Applied Mathematics, Eur. Math. Soc., Zürich, (2009), 43-56.  doi: 10.4171/056-1/3.  Google Scholar

[12]

L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

[13]

E. ChasseigneM. Chaves and J. D. Rossi, Asymptotic behaviour for nonlocal diffusion equations, J. Math. Pures Appl., 86 (2006), 271-291.  doi: 10.1016/j.matpur.2006.04.005.  Google Scholar

[14]

F. Colasuonno and P. Pucci, Multiplicity of solutions for $p(x)$-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.  doi: 10.1016/j.na.2011.05.073.  Google Scholar

[15]

C. CortazarM. ElguetaJ. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002.  Google Scholar

[16]

A. Di CastroT. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836.  doi: 10.1016/j.jfa.2014.05.023.  Google Scholar

[17]

A. Di CastroT. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003.  Google Scholar

[18]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[19]

J. M. do'OO. H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1395-1410.  doi: 10.1007/s00030-015-0327-0.  Google Scholar

[20]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.   Google Scholar

[21]

M. Fila, Boundedness of global solutions of nonlinear diffusion equations, J. Differential Equations, 98 (1992), 226-240.  doi: 10.1016/0022-0396(92)90091-Z.  Google Scholar

[22]

A. FiscellaR. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.  doi: 10.5186/aasfm.2015.4009.  Google Scholar

[23]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011.  Google Scholar

[24]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Math. Univ. Parma, 5 (2014), 373-386.   Google Scholar

[25]

M. Gobbino, Quasilinear degenerate parabolic equations of Kirchhoff type, Math. Meth. Appl. Sci., 22 (1999), 375-388.  doi: 10.1002/(SICI)1099-1476(19990325)22:5<375::AID-MMA26>3.0.CO;2-7.  Google Scholar

[26]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[27]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[28]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.  doi: 10.1007/s00526-013-0600-1.  Google Scholar

[29]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977.  doi: 10.1016/j.na.2005.03.021.  Google Scholar

[30]

X. MingqiG. Molica BisciG. H. Tian and B. L. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 357-374.  doi: 10.1088/0951-7715/29/2/357.  Google Scholar

[31]

M. Pérez-Llanosa and J. D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640.  doi: 10.1016/j.na.2008.02.076.  Google Scholar

[32]

P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.  doi: 10.4171/RMI/879.  Google Scholar

[33]

P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214.  doi: 10.1006/jdeq.1998.3477.  Google Scholar

[34]

P. PucciM. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogenous Schrodinger-Kirchhoff type equations involving the fractional $p-$Laplacian in $\mathbb{R}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5.  Google Scholar

[35]

P. PucciM. Q. Xiang and B. L. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102.  Google Scholar

[36]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs Vol. 49, American Mathematical Society, Providence, RI, 1997, xiv + 278 pp.  Google Scholar

[37]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symp., Springer, Heidelberg, 7 (2012), 271-298.  doi: 10.1007/978-3-642-25361-4_15.  Google Scholar

[38]

M. Q. XiangB. L. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041.  doi: 10.1016/j.jmaa.2014.11.055.  Google Scholar

[39]

M. Q. XiangB. L. Zhang and M. Ferrara, Multiplicity results for the nonhomogeneous fractional $p$-Kirchhoff equations with concave-convex nonlinearities, Proc. Roy. Soc. A, 471 (2015), 20150034, 14 pp.  doi: 10.1098/rspa.2015.0034.  Google Scholar

[40]

M. Q. XiangB. L. Zhang and V. Rădulescu, Existence of solutions for perturbed fractional $p$-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413.  doi: 10.1016/j.jde.2015.09.028.  Google Scholar

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